#6814838

order	512
nilpotency class	3
isoclinism	isoclinic class of #6814815
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1830 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 40 points:

(1,2)(3,7)(4,6)(5,8)(9,10,14,19)(11,20,23,31)(12,25,15,17)(13,18,30,36)(16,21,33,24)(22,37,28,40)(26,27,29,34)(32,38,35,39);(1,3)(2,5)(4,8)(6,7)(9,10)(11,20)(12,17)(13,27)(14,19)(15,25)(16,21)(18,26)(22,32)(23,31)(24,33)(28,35)(29,36)(30,34)(37,39)(38,40);(9,11,14,23)(10,16)(12,24,15,21)(13,28,30,22)(17,20)(18,35)(19,33)(25,31)(26,38,29,39)(27,37)(32,36)(34,40);(2,6)(5,7)(9,12,14,15)(10,17,19,25)(11,21,23,24)(13,29,30,26)(16,31,33,20)(18,27,36,34)(22,39,28,38)(32,37,35,40);(3,8)(5,7)(9,13,15,29,14,30,12,26)(10,18,17,34,19,36,25,27)(11,22,21,38,23,28,24,39)(16,32,20,37,33,35,31,40)

Polycyclic presentation:

2^9|a^2=i,c^2=ghi,d^2=i,e^2=h,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,h^b=hi,d^c=di,e^c=ei,e^d=ei

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6722034);